2008 AMC 12B Problem 16

Problem 16 of 25IntermediateAlgebra Geometry Number Theory

A rectangular floor measures

a

feet by

b

feet, where

a

and

b

are positive integers with

b \gt a.

An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width

1

foot around the painted rectangle and occupies half the area of the entire floor. How many possibilities are there for the ordered pair

(a, b)?

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Solution

The painted rectangle measures

(a - 2)

(b - 2)

and has half the area of the floor, so

ab = 2(a - 2)(b - 2).

Expanding gives

0 = ab - 4a - 4b + 8,

and adding

8

yields

(a - 4)(b - 4) = 8.

With

b \gt a \gt 0,

the only valid factor pairs of

8

are

(a - 4, b - 4) = (1, 8)

and

(2, 4),

giving

(a, b) = (5, 12)

and

(6, 8).

There are

2

possibilities. Thus, the correct answer is B.

Answer choices

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